Discrete distributions

A binomial experiment counts successes in a fixed number of independent trials where each trial has the same probability of success. If $X$ is the number of successes in $n$ trials with success probability $p$, then $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$ for $k=0,1,\dots ,n$. The binomial coefficient $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$ counts how many ways the $k$ successes can be arranged among the $n$ trials. For cumulative probabilities, add the individual probabilities for all values in the desired range. For example, $P(X\le 2)=P(X=0)+P(X=1)+P(X=2)$, and $P(X>3)=1-P(X\le 3)$.